Each coordinate in the plane, can be mapped onto a complex number, e.g. (1,1) maps to 1+i. A complex function is something that takes a complex number as input and spits out another complex number as output. The Mathematica program applies a complex function to each point in the complex plane and plots a color according to the argument of the number returned.
We invite you to download the mathematica notebook to try and use yourself.
Some sample plots are displayed below.
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